Determine the conditional probabilities of error

Question

Solve the following problem:

A ternary communication system transmits one of three equiprobable signals s(t), 0, or -s(t) every T seconds. The received signal is r₁(t) = s(t) + z(t), r₁(t) = z(t), or r_l(t) = -s(t) + z(t), where z(t) is white Gaussian noise with E[z(t)] = 0 and R_z(τ ) = E [z(t)z∗(τ )] = 2N₀δ(t - τ ).

The optimum receiver computes the correlation metric

U = Re [ ∫₀^T  r_l(t)s*(t)dt]

and compares U with a threshold A and a threshold -A. If U > A, the decision is made that s(t) was sent. If U

1. Determine the three conditional probabilities of error: Pe given that s(t) was sent, Pe given that -s(t) was sent, and Pe given that 0 was sent.

2. Determine the average probability of error Pe as a function of the threshold A, assuming that the three symbols are equally probable a priori.

3. Determine the value of A that minimizes Pe.